For the Cantor-styled artificial DNA sequence constructed by following the inflation rule A→ABA and B→BBB, using the one-dimensional random walk model and Hurst analysis, we calculate numerically the displacement and its standard deviation, the rescaled range function and the Hurst exponent. The results are compared with those of one-dimensional random binary sequence. We show that Cantor-styled DNA sequence presents correlated behavior, scaling invariability, and self-similarity. In addition, according to the renormalization-group method, we study the charge transfer properties of Cantor-styled DNA sequence, including localization length, charge transmission coefficient and Lyapunov exponent. We find that some resonant peaks can survive in relatively long Cantor-styled DNA sequences, which also implies that there are long-range correlations in Cantor-styled DNA sequences.